naginterfaces.library.specfun.tanh

naginterfaces.library.specfun.tanh(x)

tanh returns a value for the hyperbolic tangent, $\tanh \left(x\right)$.

For full information please refer to the NAG Library document for s10aa

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/s/s10aaf.html

Parameters

xfloat

The argument $x$ of the function.

Returns

thxfloat

The value of the hyperbolic tangent, $\tanh \left(x\right)$.

Notes

tanh calculates an approximate value for the hyperbolic tangent of its argument, $\tanh \left(x\right)$.

For $\left|x\right|\le 1$ it is based on the Chebyshev expansion

$$\tanh \left(x\right)=x\times y\left(t\right)=x{\sum ^{\prime }}_{r=0}{a}_r{T}_r\left(t\right)$$

where $-1\le x\le 1\text{,}-1\le t\le 1\text{, and}t=2x^2-1$.

For $1<\left|x\right|<E_1$

$$\tanh \left(x\right)=\frac{e^{2x}-1}{e^{2x}+1}\text{.}$$

For $\left|x\right|\ge E_1$, $\tanh \left(x\right)=sign\left(x\right)$ to within the representation accuracy of the machine and so this approximation is used.

References

NIST Digital Library of Mathematical Functions